Periodic points on the $2$-sphere
Charles Pugh Michael Shub
Discrete & Continuous Dynamical Systems - A 2014, 34(3): 1171-1182 doi: 10.3934/dcds.2014.34.1171
For a $C^{1}$ degree two latitude preserving endomorphism $f$ of the $2$-sphere, we show that for each $n$, $f$ has at least $2^{n}$ periodic points of period $n$.
keywords: latitude preserving. Periodic points smoothness two sphere degree two
Entropy estimates for a family of expanding maps of the circle
Rafael De La Llave Michael Shub Carles Simó
Discrete & Continuous Dynamical Systems - B 2008, 10(2&3, September): 597-608 doi: 10.3934/dcdsb.2008.10.597
In this paper we consider the family of circle maps $f_{k,\alpha,\epsilon}:\mathbb{S}^1\rightarrow \mathbb{S}^1$ which when written mod 1 are of the form $f_{k,\alpha,\epsilon}: x \mapsto k x + \alpha + \epsilon \sin(2\pi x)$, where the parameter $\alpha$ ranges in $\mathbb{S}^1$ and $k\geq 2.$ We prove that for small $\epsilon$ the average over $\alpha$ of the entropy of $f_{k,\alpha,\epsilon}$ with respect to the natural absolutely continuous measure is smaller than $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx,$ while the maximum with respect to $\alpha$ is larger. In the case of the average the difference is of order of $\epsilon^{2k+2}.$ This result is in contrast to families of expanding Blaschke products depending on rotations where the averages are equal and for which the inequality for averages goes in the other direction when the expanding property does not hold, see [4]. A striking fact for both results is that the maximum of the entropies is greater than or equal to $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx$. These results should also be compared with [3], where similar questions are considered for a family of diffeomorphisms of the two sphere.
keywords: entropy estimates expanding circle maps.
Errata to "Stably ergodic skew products"
Roy Adler Bruce Kitchens Michael Shub
Discrete & Continuous Dynamical Systems - A 1999, 5(2): 456-456 doi: 10.3934/dcds.1999.5.456
keywords: Measure preserving Anosov diffeomorphism ergodic skew products
Stably ergodic skew products
Roy Adler Bruce Kitchens Michael Shub
Discrete & Continuous Dynamical Systems - A 1996, 2(3): 349-350 doi: 10.3934/dcds.1996.2.349
In [PS] it is conjectured that among the volume preserving $C^2$ diffeomorphisms of a closed manifold which have some hyperbolicity, the ergodic ones contain an open and dense set. In this paper we prove an analogous statement for skew products of Anosov diffeomorphisms of tori and circle rotations. Thus this paper may be seen as an example of the phenomenon conjectured in [PS]. The corresponding theorem for skew products of Anosov diffeomorphisms and translations of arbitrary compact groups is an interesting open problem.
keywords: skew products compact groups.
Hölder foliations, revisited
Charles Pugh Michael Shub Amie Wilkinson
Journal of Modern Dynamics 2012, 6(1): 79-120 doi: 10.3934/jmd.2012.6.79
We investigate transverse Hölder regularity of some canonical leaf conjugacies in normally hyperbolic dynamical systems and transverse Hölder regularity of some invariant foliations. Our results validate claims made elsewhere in the literature.
keywords: normal hyperbolicity invariant foliations H\"older regularity of foliations. Partial hyperbolicity
Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms
Charles Pugh Michael Shub Alexander Starkov
Discrete & Continuous Dynamical Systems - A 2006, 14(4): 845-855 doi: 10.3934/dcds.2006.14.845
In 1954, F. Mautner gave a simple representation theoretic argument that for compact surfaces of constant negative curvature, invariance of a function along the geodesic flow implies invariance along the horocycle flows (these are facts which imply ergodicity of the geodesic flow itself), [M]. Many generalizations of this Mautner phenomenon exist in representation theory, [St1]. Here, we establish a new generalization, Theorem 2.1, whose novelty is mostly its method of proof, namely the Anosov-Hopf ergodicity argument from dynamical systems. Using some structural properties of Lie groups, we also show that stable ergodicity is equivalent to the unique ergodicity of the strong stable manifold foliations in the context of affine diffeomorphisms.
keywords: manifold foliations affine diffeomorphisms. stable ergodicity Representation theory the Anosov-Hopf ergodicity Lie groups unique ergodicity

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